Abstract
The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of thenth order delay differential equations(E)(r(t)[x(n−1)(t)]γ)′+q(t)xγ(τ(t))=0. The results obtained utilize also the comparison theorems.
Highlights
As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the nth order delay differential equations Ertxn−1 t γ q t xγ τ t
Throughout the paper, we will assume q, τ, r ∈ C t0, ∞, and H1 n ≥ 3, γ is the ratio of two positive odd integers, H2 r t > 0, q t > 0, τ t ≤ t, limt → ∞τ t ∞
The problem of the oscillation of higher-order differential equations has been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for studied equations see, e.g., 1–19
Summary
We will study the asymptotic and oscillation behavior of the solutions of the higher-order advanced differential equations:r t x n−1 t γ q t xγ τ t EThroughout the paper, we will assume q, τ, r ∈ C t0, ∞ , and H1 n ≥ 3, γ is the ratio of two positive odd integers, H2 r t > 0, q t > 0, τ t ≤ t, limt → ∞τ t ∞.Whenever, it is assumed tRt r−1/γ s ds −→ ∞ as t −→ ∞. t0International Journal of Mathematics and Mathematical SciencesBy a solution of E we mean a function x t ∈ Cn−1 Tx, ∞ , Tx ≥ t0, which has the property r t x n−1 t γ ∈ C1 Tx, ∞ and satisfies E on Tx, ∞. The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the nth order delay differential equations Ertxn−1 t γ q t xγ τ t We will study the asymptotic and oscillation behavior of the solutions of the higher-order advanced differential equations: r t x n−1 t γ q t xγ τ t
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